A universal diffuse interface modeling framework for surfactants

Abstract

We propose a universal diffuse-interface modeling framework for surfactant transport in two-phase flows, applicable to all solubility scenarios and to any conservative phase field method. The foundation of our framework is a general three-scalar non-equilibrium model governing the surfactant concentrations in each bulk phase and at the interface, which builds on our prior consistent scalar transport framework and is locally and globally conservative, leakage-free, Galilean-invariant, and reduction-consistent. Assuming thermochemical equilibrium, we derive two one-scalar models: one for a surfactant in full equilibrium between both bulk phases and the interface, and one for a surfactant confined to a single bulk phase and the interface. All models are coupled to the Navier-Stokes equations through a surface tension force that incorporates the Marangoni stress arising from non-uniform interfacial surfactant distributions. While diffuse-interface surfactant models have been developed for the Cahn-Hilliard equation and, more recently, for the conservative Allen-Cahn (CAC) equation, existing models for CAC address only the insoluble and single-phase-soluble cases, leaving the general scenario of partial solubility in both bulk phases unaddressed; furthermore, these models lack Galilean invariance and reduction consistency, and are not applicable beyond the CAC setting. The framework is validated against analytical solutions in one-dimensional transport tests, assessed for convergence in two-dimensional advection-diffusion simulations, and demonstrated in fully-coupled drop-in-shear flow simulations covering insoluble, soluble, and partially-soluble surfactant scenarios.